Intuitive explanation of topological genus?

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I see that the standard definition of genus is:

"A topologically invariant property of a surface defined as the largest number of nonintersecting simple closed curves that can be drawn on the surface without separating it. Roughly speaking, it is the number of holes in a surface. "

Why is it that under this definition, a sphere has a genus of 0, while a torus has a genus of 1?

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1 Answer

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"Why is it that under this definition, a sphere has a genus of 0, while a torus has a genus of 1?"

Intuitively speaking, every closed curve in $S^2$ separate it in at least two pieces (think about the equator or a parallel or something like this).

The torus is different. There exist (at least one) closed curve that do not separate it. Think about curves like these:

These curves (individually) separate torus on a connected topological space (a cylinder or a circular crown). And both cylinder and circular crown have genus 0 (why?). Thus, torus has genus 1.

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